When is the best time of year within the UK to dig and transfer a pear tree to another garden ?

When the tree is dormant usually between November and March depending whether it's an early or late flowering variety. You probably still have another couple of weeks in which to transplant the tree. Make sure that when you dig it up you get as much of the rootball and soil around as you can and if possible have the planting hole ready dug for it in it's new home and plant it straight away. If that is not possible wrap the roots in sacking and put in a cool frost free place until you can. When it starts to sprout make sure to keep it well watered. You may find you won't get much fruit this year.

trees are funny. I'd have a nurseryman do it.

you have left it a bit late for this year ...you just might get lucky if you do it TODAY ..but mine is showing signs of buds ...November -January is best

When it's not raining. So never.

When dormant........before the buds begin to swell.